Spontaneous Breaking of Classical PT Symmetry

The classical trajectories of the family of complex PT-symmetric Hamiltonians $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) form closed orbits. All such complex orbits that have been studied in the past are PT symmetric (left-right symmetric). The periods of these orbits exhibit an unusual dependence on the parameter $\epsilon$. There are regions in $\epsilon$ of smooth behavior interspersed with regions of rapid variation. It is demonstrated that the onset of rapid variation is associated with strange new kinds of classical trajectories that have never been seen previously. These rare kinds of trajectories are not PT symmetric and occur only for special rational values of $\epsilon$.